Characterizing distinction by self-duality for l = 2 across involutions and inner forms

Determine whether, for any inner form G of GL_n(F) and any involution θ of G, a cuspidal irreducible F̄_2-representation π of G is H-distinguished (with H = G^θ) if and only if π is θ-self-dual, i.e., π^∨ ≅ π ∘ θ.

Background

Theorem 1.3 shows that, for GL_n(F) with the Galois involution coming from a quadratic extension F/F_0 and l = 2, a cuspidal F̄_2-representation is H-distinguished precisely when it is conjugate-self-dual.

The authors suggest that an analogous statement should hold for any involution on any inner form of GL_n(F): distinction might coincide with θ-self-duality in characteristic 2. Establishing this would extend the neat self-duality criterion from the Galois case to a much broader class of symmetric pairs.

References

It is tempting to conjecture that Theorem 1.2 holds in greater generality. For other involutions of the group GLn (F), or more generally for involutions of inner forms of GLn (F), there is some hope indeed. It seems plausible as well that a result similar to Theorem 1.3 holds for any involution of any inner form of GLn (F).

— Cuspidal $\ell$-modular representations of ${\rm GL}_n(F)$ distinguished by a Galois involution, II  (2604.01931 - Kurinczuk et al., 2 Apr 2026) in Section 1.10